📚 Functions: Domain and Range | 函数的定义域与值域
In IGCSE Edexcel Mathematics, understanding the domain and range of a function is essential for solving problems involving graphs, composite functions, and inverse functions. This article will guide you through the definitions, notation, and practical methods to determine both domain and range, with worked examples tailored to the exam style.
在 Edexcel IGCSE 数学中，理解函数的定义域与值域是解决图像、复合函数和反函数问题的关键。本文将带你系统地掌握两者的定义、表示方法以及求解技巧，并结合考试常见题型给出典型例题。
1. What is a Function? | 什么是函数？
e range). It is often written as f(x) = ... , where x is the input and f(x) is the output.
函数是一种对应规则：它将定义域中的每一个输入值，唯一地对应到值域中的一个输出值。通常写作 f(x) = ...，其中 x 是输入，f(x) 是输出。
3, the output is f(3) = 2(3) + 1 = 7. Every x gives exactly one y, so this is a function.
例如，f(x) = 2x + 1 表示如果你输入 x = 3，输出为 f(3) = 2(3) + 1 = 7。每个 x 都对应唯一的 y，因此这是一个函数。
A simple vertical line test can be used on a graph: if any vertical line crosses the graph more than once, the graph does not represent a function.
在图像上可以用“垂直线检验法”判断：若任何一条竖直线与图像相交超过一次，则该图像不表示一个函数。
2. Defining the Domain | 定义域的含义
The domain of a function is the complete set of possible input values (x-values) for which the function is defined. In IGCSE, the domain is often given explicitly, but sometimes you must identify it from the context or the formula.
定义域是函数中所有可能的输入值（即 x 值）的集合，在这些值上函数有意义。在 IGCSE 中，定义域有时会直接给出，但有时也需要你自己根据公式或实际情境判断。
Common restrictions on the domain include:
定义域常见的限制条件包括：
- Denominators cannot be zero — for example, in f(x) = 1/(x-2), x ≠ 2.
- 分母不能为零——例如 f(x) = 1/(x-2)，则 x ≠ 2。
- Expressions under a square root must be non-negative — f(x) = √x-3 requires x ≥ 3.
- 偶次根号下的表达式必须非负——f(x) = √x-3 要求 x ≥ 3。
- In word problems, only sensible values are allowed (e.g., time cannot be negative).
- 实际应用题中只取合理值（例如时间不能为负数）。
ys write the domain using set notation or interval notation, such as {x : x ≠ 2} or x ≥ 3.
书写定义域时，应使用集合符号或区间符号，例如 {x : x ≠ 2} 或 x ≥ 3。
3. Defining the Range | 值域的含义
The range of a function is the complete set of all possible output values (y-values) that the function can produce. It depends on both the formula and the domain.
值域是函数所有可能的输出值（即 y 值）的集合。它由函数的表达式和定义域共同决定。
To find the range, you ask: "What y-values can f(x) actually take when x runs over the domain?" For example, f(x) = x² with domain x ∈ ℝ has range y ≥ 0 (since squares are never negative).
求值域时，你只需问：“当 x 在定义域内变化时，f(x) 究竟能取到哪些 y 值？”例如，f(x) = x²，定义域为全体实数时，值域为 y ≥ 0（因为平方永远不可能为负）。
f(x) = x² with domain 0 ≤ x ≤ 3, the outputs range from 0 to 9, so the range is 0 ≤ y ≤ 9.
如果定义域受到限制，值域也会随之改变。对于 f(x) = x²，定义域为 0 ≤ x ≤ 3 时，输出值从 0 到 9，因此值域为 0 ≤ y ≤ 9。
4. Finding the Range from a Graph | 从图像上找值域
The easiest way to find the range is to sketch the graph, then project the curve onto the y-axis. The visible portion of the y-axis tells you the range.
找值域最简单的方法是先画出函数图像，然后将曲线投影到 y 轴上。y 轴上被覆盖的部分就是值域。
pens upwards, the lowest output is -4 and there is no upper limit, so the range is y ≥ -4.
考虑 f(x) = x² - 4。图像是一条开口向上的抛物线，顶点在 (0, -4)。抛物线的最低输出值是 -4，没有上限，因此值域为 y ≥ -4。
0) = -4, f(2) = 0. The smallest value is -4, the largest is 0, so the range is -4 ≤ y ≤ 0.
当定义域受限时，你只需要考虑定义域内的 x 值。对于 f(x) = x² - 4，定义域为 -1 ≤ x ≤ 2 时，检验端点：f(-1) = -3，f(0) = -4，f(2) = 0。最小值为 -4，最大值为 0，因此值域为 -4 ≤ y ≤ 0。
5. Domain and Range of Linear Functions | 线性函数的定义域与值域
he domain is all real numbers, then the range is also all real numbers. The graph is a straight line that extends infinitely in both directions.
对于线性函数 f(x) = mx + c，若定义域为全体实数，则值域也是全体实数。图像是一条向两端无限延伸的直线。
5, then f(1) = 3 and f(5) = 11. Since the function is increasing, the range is 3 ≤ y ≤ 11.
如果定义域受限，值域也相应受限。对于 f(x) = 2x + 1，定义域为 1 ≤ x ≤ 5，则 f(1) = 3，f(5) = 11。由于函数单调递增，值域为 3 ≤ y ≤ 11。
r f(x) = 3 - 2x with domain 0 ≤ x ≤ 4, f(0) = 3 and f(4) = -5, so the range is -5 ≤ y ≤ 3.
对于递减函数，顺序相反：最大的 y 来自最小的 x。对于 f(x) = 3 - 2x，定义域为 0 ≤ x ≤ 4，f(0) = 3，f(4) = -5，因此值域为 -5 ≤ y ≤ 3。
6. Domain and Range of Quadratic Functions | 二次函数的定义域与值域
have a parabolic graph. The turning point determines either the maximum or the minimum value of the range.
二次函数 f(x) = ax² + bx + c 的图像为抛物线。其顶点决定了值域的最大值或最小值。
If a > 0, the parabola opens upwards, and the vertex gives the minimum value of the range. Complete the square to find the vertex:
若 a > 0，抛物线开口向上，顶点给出值域的最小值。通过配方可以找到顶点：
f(x) = a(x-h)² + k
where (h, k) is the vertex. The range is y ≥ k (if a > 0) or y ≤ k (if a < 0).
其中 (h, k) 是顶点坐标。值域为 y ≥ k（若 a > 0）或 y ≤ k（若 a < 0）。
Worked example: Find the range of f(x) = x² - 6x + 10, with domain x ∈ ℝ.
典型例题：求 f(x) = x² - 6x + 10 在定义域为全体实数时的值域。
Complete the square:
配方：
x² - 6x + 10 = (x-3)² + 1
. Since the coefficient of x² is positive, the minimum output is 1, so the range is y ≥ 1.
顶点为 (3, 1)。由于 x² 系数为正，最小输出值为 1，因此值域为 y ≥ 1。
7. Domain of Functions with Square Roots | 含平方根函数的定义域
e expression inside the root must be greater than or equal to zero. This condition gives you the domain.
对于含有 √(g(x)) 的函数，根号内的表达式必须大于或等于零。由这个条件可以确定定义域。
Worked example: Find the domain of f(x) = √2x - 6.
典型例题：求 f(x) = √2x - 6 的定义域。
Set the radicand ≥ 0:
令根号内表达式 ≥ 0：
2x - 6 ≥ 0
2x ≥ 6
x ≥ 3
So the domain is {x : x ≥ 3}. To find the range, note that √2x - 6 starts at 0 when x = 3 and grows without bound, so the range is y ≥ 0.
因此定义域为 {x : x ≥ 3}。求值域时，注意到当 x = 3 时 √2x - 6 从 0 开始，且无限增大，因此值域为 y ≥ 0。
If the square root appears in the denominator, the radicand must be strictly positive, not just non-negative, because division by zero is undefined.
如果平方根出现在分母中，那么根号内的表达式必须严格大于零，而不是仅仅非负，因为除以零是无意义的。
8. Domain of Functions with Fractions | 含分式函数的定义域
When a function has a denominator, you must exclude any x-values that make the denominator equal to zero.
当函数含有分母时，必须排除所有使分母为零的 x 值。
Worked example: Find the domain of f(x) = 3/(x - 5).
典型例题：求 f(x) = 3/(x - 5) 的定义域。
Set the denominator ≠ 0:
令分母 ≠ 0：
x - 5 ≠ 0
x ≠ 5
So the domain is {x : x ≠ 5}. In interval notation, this is (-∞, 5) ∪ (5, ∞).
因此定义域为 {x : x ≠ 5}。用区间表示即为 (-∞, 5) ∪ (5, ∞)。
- 4), set x² - 4 ≠ 0, so x ≠ 2 and x ≠ -2. The domain is all real numbers except 2 and -2.
对于 f(x) = 1/(x² - 4)，令 x² - 4 ≠ 0，得 x ≠ 2 且 x ≠ -2。定义域为除 2 和 -2 以外的所有实数。
For f(x) = 1/(x-5), as x → ∞, f(x) → 0, but f(x) never equals 0. Thus the range is y ≠ 0.
求分式函数的值域通常需要考虑水平渐近线。对于 f(x) = 1/(x-5)，当 x → ∞ 时，f(x) → 0，但永远不会等于 0。因此值域为 y ≠ 0。
9. Using Set Notation and Interval Notation | 集合符号与区间符号的运用
In IGCSE exams, you may use either set notation or interval notation to express domains and ranges. Familiarity with both is essential.
在 IGCSE 考试中，你可以使用集合符号或区间符号来表达定义域和值域。熟练掌握两种表示方法非常重要。
Common forms include:
常见形式包括：
- Set notation: {x : x ≥ 3}, {x : x ≠ -2}
- 集合符号：{x : x ≥ 3}，{x : x ≠ -2}
- Interval notation: [3, ∞), (-∞, -2) ∪ (-2, ∞)
- 区间符号：[3, ∞)，(-∞, -2) ∪ (-2, ∞)
- Inequality notation: x ≥ 3, -4 ≤ y ≤ 0
- 不等式符号：x ≥ 3，-4 ≤ y ≤ 0
d brackets ( ) exclude it. ∞ always uses a round bracket because infinity is not a number.
请记住，方括号 [ ] 表示包含端点，圆括号 ( ) 表示不包含端点。∞ 前永远使用圆括号，因为无穷大不是一个具体数值。
For domain and range, always pay attention to whether endpoints are included. A square-root function includes its minimum, but a reciprocal function excludes both its vertical and horizontal asymptote values.
在写定义域和值域时，务必注意端点是否包含。平方根函数包含其最小值，但分式函数既不包含垂直渐近线上的值，也不包含水平渐近线的值。
10. Composite Functions: How the Domain Changes | 复合函数：定义域如何变化
When you form a composite function f(g(x)), the domain must satisfy two conditions: x must be in the domain of g, and g(x) must be in the domain of f. This is a common source of errors in exams.
构造复合函数 f(g(x)) 时，定义域必须满足两个条件：x 必须在 g 的定义域内，且 g(x) 必须在 f 的定义域内。这是考试中常见的易错点。
Worked example: Let f(x) = √x and g(x) = 2x + 3. Find the domain of f(g(x)).
典型例题：已知 f(x) = √x，g(x) = 2x + 3。求 f(g(x)) 的定义域。
First, f(g(x)) = √2x + 3. For the square root to be defined, we need:
首先，f(g(x)) = √2x + 3。要使根号有意义，我们需要：
2x + 3 ≥ 0
x ≥ -3/2
bers, so the overall domain is x ≥ -3/2. There is no additional restriction from g itself.
g 的定义域为全体实数，因此复合函数的定义域为 x ≥ -3/2。g 本身没有额外的限制。
s excluded. For example, if f(x) = 1/x and g(x) = x - 2, then f(g(x)) = 1/(x-2), so x ≠ 2.
如果 g 的某些输出值不在 f 的定义域内，相应的 x 也要被排除。例如，若 f(x) = 1/x，g(x) = x - 2，则 f(g(x)) = 1/(x-2)，因此 x ≠ 2。
11. Inverse Functions and Their Domains | 反函数及其定义域
Every one-to-one function has an inverse. The key relationship is:
每一个一一对应的函数都有反函数。关键关系是：
Domain of f⁻¹ = Range of f
Range of f⁻¹ = Domain of f
f⁻¹ 的定义域 = f 的值域
f⁻¹ 的值域 = f 的定义域
Worked example: Let f(x) = 2x + 3 with domain 0 ≤ x ≤ 4. Find the domain and range of f⁻¹.
典型例题：已知 f(x) = 2x + 3，定义域为 0 ≤ x ≤ 4。求 f⁻¹ 的定义域和值域。
ange of f. Since f is increasing, f(0) = 3 and f(4) = 11, so the range of f is 3 ≤ y ≤ 11.
先求 f 的值域。由于 f 单调递增，f(0) = 3，f(4) = 11，所以 f 的值域为 3 ≤ y ≤ 11。
f⁻¹ is 3 ≤ x ≤ 11, and range of f⁻¹ is 0 ≤ y ≤ 4. This is consistent with the rule above.
对于反函数：f⁻¹ 的定义域为 3 ≤ x ≤ 11，f⁻¹ 的值域为 0 ≤ y ≤ 4。这与上述规则完全一致。
y = (x-3)/2. The domain restriction 0 ≤ x ≤ 4 for f becomes the range restriction for f⁻¹.
求反函数公式时，交换 x 和 y 并重新整理：y = 2x + 3 变为 x = 2y + 3，因此 y = (x-3)/2。f 的定义域 0 ≤ x ≤ 4 变成了 f⁻¹ 的值域限制。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Here are the most frequent pitfalls students encounter in the exam, and how to avoid them:
以下是同学们在考试中最常遇到的问题以及避免方法：
- Forgetting that a square root requires non-negative radicand.
- 忘记平方根要求根号内非负。
- Dividing by zero: always exclude values that make the denominator zero.
- 除以零：务必排除使分母为零的 x 值。
- Assuming the range of a quadratic is all real numbers — it is never all real numbers.
- 误以为二次函数的值域是全体实数——实际上永远不是。
- Forgetting to check whether the endpoints are included in the range.
- 忘记检查值域的端点是否包含。
- In composite functions, forgetting to restrict the domain of the inner function first.
- 在复合函数中，忘记先限制内层函数的定义域。
- When finding the inverse, forgetting to swap the domain and range restrictions.
- 求反函数时，忘记交换定义域和值域的限制。
Always sketch the graph when possible. A clear sketch helps you instantly see the domain and range, especially for quadratic, reciprocal, and square-root functions.
在可能的情况下，一定要画出函数图像。一幅清晰的草图能帮助你直观地看出定义域和值域，尤其对于二次、分式和平方根函数非常有效。
Finally, always write your final answer in the notation requested by the question. If the question says "use set notation", do not use interval notation in your answer.
最后，务必按照题目要求的符号书写答案。如果题目要求“用集合符号表示”，就不要用区间符号作答。
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